9-16_Elasticity_tutorial

# 9-16_Elasticity_tutorial - Silicon as an anisotropic...

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Silicon as an anisotropic mechanical material - a tutorial Ville Kaajakari This tutorial covers the calculation of silicon Young’s modulus and Poisson’s ratio from elastic constants in any crystal orientation. The algebra is the same for any elastic material with cubic symmetry but I am mostly interested in silicon as I have used it to make micromechanical compo- nents. The tutorial assumes knowledge of matrix algebra and some elementary mechanics concepts such as stress and strain. The material in this tutorial is mainly based on the paper by Wortman and Evans [ 1 ] with some concepts not familiar for a typical engineer brieﬂy explained. Figure 1 below shows how Young’s modulus Y is deﬁned: the bar is stretched in the x -direction while simultaneously it is allowed to move freely in y - and z -directions. The Young’s modulus is then deﬁned as the ratio of stress to strain in the direction of the stretching ( Y = T 11 / ε 11 ). The Pois- son’s ratio is deﬁned as ratio of length extension to sideways contraction ( ν = - ε 22 / ε 11 ). Different directions are referred with numbers and letters interchangeably with numbers 1, 2, and 3 used to indicate x , y , and z axes respectively. A F T 1 11 = 0 11 l l x u x = = ε Y slope 11 T 0 33 22 = = T T 1 F 0 3 2 = = F F undeformed shape deformed shape x y z Figure 1. Deﬁnition of Young’s modulus Y. This tutorial uses numbers 1 , 2 , and 3 to indicate x, y, and z axes respectively. For an anisotropic material such as silicon the Young’s modulus depends on which crystal di- rection the material is being stretched. Looking at Figure 2 this should be no surprise as the silicon Copyright Ville Kaajakari ([email protected]) Homepage: http://www.kaajakari.net Tutorials: http://www.kaajakari.net/ ~ ville/research/tutorials/tutorials.shtml 1

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crystal is highly structured. Figure 2 is also a quick introduction to the crystallographic notation: Different directions are indicated with respect to crystal basis using Miller indexes. In cubic crystal such as silicon the [100], [010] and [001]-directions can be chosen to coincide with x , y , and z -axes. However, this may not be true for crystal with different symmetry. The Miller indexes can be thought
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9-16_Elasticity_tutorial - Silicon as an anisotropic...

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